# Algebra, Geometry, and Cryptology Seminar

## Fall 2019

Organized by Zach Teitler <zteitler@boisestate.edu>.

Time: Fridays, 3:00-3:50
Location: MB124

View archives of past semesters

• Taking the seminar for credit
The seminar may be taken for credit as Math 498 or 598. For information about receiving credit, contact zteitler@boisestate.edu.
• Everybody is welcome!
Everybody interested is welcome to attend and participate! Enrollment for credit is not required. We welcome students to attend and present in the seminar. Attending seminars and colloquium presentations is an excellent way to learn about research topics and senior thesis topics.
• AGC Seminar mailing list
Announcements of upcoming seminars are sent by email to all interested participants. To be added to the AGC Seminar mailing list, contact zteitler@boisestate.edu.

Planning meeting

### September 6

Topology Seminar

Allison Arnold-Roksandich, BSU
Normal Mathematics

Kennedy Courtney, BSU
Index and the Poincare-Hopf Theorem

### September 13

Allison Arnold-Roksandich, BSU
An Introduction to Modular Forms and eta-theta Functions

Simply put, “modular forms are everywhere.” Their uses go from the abstract and theoretical (like Fermat’s Last Theorem), to encoding data (when looking at partitions), to the physical (like the study of black holes). Their presence is usually indicative of a deep underlying symmetry. This talk aims to introduce modular forms, two primary examples: Dedekind’s eta function and theta functions, and some fun results.

Slides

### September 20

Tommaso de Fernex, University of Utah
A simplicity criterion for normal isolated singularities

Let $X$ be a complex variety defined in $\mathbb{C}^n$ by the vanishing of one (or more) holomorphic functions $f(z_1,…z_n)=0$, and let $P$ be a point of $X$. Assume that $X$ is smooth (i.e., a manifold) in a punctured neighborhood of $P$; $X$ is however allowed to be singular at $P$, so that this point is an isolated singularity of $X$.  The intersection of $X$ with the boundary of a small ball in $\mathbb{C}^n$ centered at $P$ is a real hypersurface in $X$ and is called the link of $P$. It is natural to ask how much information the link carries about the singularity; for instance, the link of a smooth point is a sphere, and one can ask whether the converse is true. Work of Mumford and Brieskorn has shown that this is the case for normal surface singularities but not in higher dimensions. Recently, McLean asked whether more structure on the link may provide a way to characterize smooth points. In this talk, I will give a general introduction to this circle of questions and discuss how CR geometry can be used to distinguish smooth points from their links. The proof relies on a partial solution to the complex Plateau problem, which is a complex analytic analogue of the classical Plateau problem of Lagrange.

### September 27

Zach Teitler, BSU
Huang’s proof of the Sensitivity Conjecture

Huang has presented an elementary proof of the almost 30-year old Sensitivity Conjecture. We will introduce the conjecture and its graph-theoretic proof.

### October 4

Allison Arnold-Roksandich, BSU
An introduction to Mock Modular Forms with some examples arising from eta-theta functions

Slides

### October 11

Zach Teitler, BSU
Lucas’s theorem and MathOverflow 10

1. Lucas’s theorem on binomial coefficients.
2. The 10th anniversary of MathOverflow.

### October 18

Topology Seminar
Uwe Kaiser, BSU
Roots of the Alexander polynomial and orderability of 3-manifold groups

### October 25

Stefan Tohaneanu, University of Idaho
Projective schemes derived from linear codes

In 1999, De Boer and Pellikaan described a method to finding the minimum distance of any linear code, by the means of deciding if the heights of certain ideals are maximal or not. These ideals I have been calling “ideals generated by $a$-fold products of linear forms”, and they define some projective schemes, called “generalized star configuration schemes”. The homological/commutative algebraic properties of these ideals interact very well with the parameters of the linear code (especially with the generalized Hamming weights), and in this presentation I will survey some of these interactions and results. By taking the radicals of these ideals one can define some projective varieties that are subspace arrangements (i.e., union of linear subspaces). If time permits, I will also talk about the reverse construction: given a subspace arrangement, is it possible to construct a linear code whose associated projective variety is the given subspace arrangement. The answer is affirmative, and the question becomes about how “good” is this linear code.

### November 1

Topology Seminar
Jens Harlander, BSU
Locally indicable groups

### November 8

Allison Arnold-Roksandich, BSU
An Introduction to Quantum Modular Forms and a Few Families Arising From eta-theta Functions

In 2013, Lemke Oliver created a list of all eta-quotients which are theta functions. Then in 2016, Folsom, Garthwaite, Kang, Swisher, and Treneer utilized this list of “eta-theta” functions along with Zwegers’s construction of mock theta functions to create a set of mock modular forms which are also quantum modular forms. Later in 2016, Diaz, Ellefsen, and Swisher generalized a subset of these quantum modular forms to a single general form of quantum modular forms which included the every element of this subset. This talk will discuss an extended generalization to a larger general form for quantum modularity which encompasses all functions made by Folsom et al.

### November 15

Topology Seminar
Kennedy Courtney
, BSU
The Directed Forest Complex and Shellability

### November 22

Topology Seminar
Uwe Kaiser, BSU
Quantum Modular Forms and Knot Invariants

### November 29

No meeting this week (Thanksgiving holiday)

TBA

### December 13

No meeting this week (last week of classes)